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Copyright American Statistical Association Feb 2000
[Headnote]
The Wilcoxon-Mann-Whitney test enjoys great popularity among scientists comparing two groups of observations,
especially when measurements made on a continuous scale are non-normally distributed. Triggered by different results for
the procedure from two statistics programs, we compared the outcomes from 11 PC-based statistics packages. The
findings were that the delivered p values ranged from significant to nonsignificant at the 5% level, depending on whether a
large-sample approximation or an exact permutation form of the test was used and, in the former case, whether or not a
correction for continuity was used and whether or not a correction for ties was made. Some packages also produced
pseudo-exact p values, based on the null distribution under the assumption of no ties. A further crucial point is that the
variant of the algorithm used for computation by the packages is rarely indicated in the output or documented in the Help
facility and the manuals. We conclude that the only accurate form of the Wilcoxon-MannWhitney procedure is one in which
the exact permutation null distribution is compiled for the actual data.
KEY WORDS: Asymptotic; Continuity correction; Exact permutation test; Statistical software; Ties.
1. INTRODUCTION
The Wilcoxon-Mann-Whitney (WMW) test is very popular in the applied sciences,
especially in the life and social sciences, and specifically in the biomedical sciences
(Ludbrook and Dudley 1998). It is frequently used as the nonparametric analog of the
Student's t test to compare two sets of observations measured on an interval scale
when it is supposed that the data are non-normally distributed. It is also used, especially
in the social sciences, when the original measurements were made on an ordinal scale.
There have been reviews of how accurately microcomputer statistics packages perform
the WMW test (Bernhard, Alle, Herbold, Meyers 1988; Ludbrook 1995).
The purpose of this review is to report our experience of using 11 commercial statistical
packages to execute the Wilcoxon-Mann-Whitney test on a genuine set of experimental
data. We focus on PC-based software packages for MS-Windows, which are mainly
menu driven and widely used by applied scientists from many disciplines. The starting
point was a real dataset from a pharmacological experiment with a well-established
paradigm. When we discovered that two commercial statistical packages gave very
different outcomes from the Wilcoxon-Mann-Whitney test, we sought an explanation.
This led to our examining other statistical software packages.
2. THE WILCOXON-MANN-WHITNEY TEST
The literature refers to equivalent tests, formulated in different ways, as the Wilcoxon
rank-sum test and the MannWhitney U test. These were developed independently by
Wilcoxon (1945) and Mann and Whitney (1947). A detailed theoretical treatment of the
tests) was given by Lehmann (1998). The Wilcoxon rank-sum procedure, including
formulations of its variants, was described by Siegel and Castellan (1988) in terms that
are comprehensible to nonstatisticians. We have adopted the widely used convention of
combining the two versions as the Wilcoxon-MannWhitney (WMW) test. It is worth
noting that the majority of texts on nonparametric statistics favor the Wilcoxon version of
the test, in which the rank-sums of the groups are calculated, over the Mann-Whitney
version in which a U statistic is used.
2.1 Nature of the Hypothesis Tested by the WMW Procedure
This depends on whether one adheres to the classical, population, model of inference,
or to the randomization model. This, in turn, depends on the sampling technique
employed.
Under the classical (population) model the null hypothesis is that the two populations
have the same response distribution against the alternative that they are different. The
WMW test is used to detect "shifted alternatives". That is, the two population
distributions have the same general shape (including dispersions), but one of them is
shifted relative to the other by a constant amount under the alternative hypothesis. The
statistical inference is generalizable to future, similar experiments in which random
sampling is employed.
Under the randomization model, which is the norm in biomedical research (Ludbrook
and Dudley 1998), the WMW procedure tests specifically whether there is a difference
between randomized groups in terms of their mean ranks. The statistical inference
applies only to the actual experiment performed.
Whichever model of inference is appropriate to the experimental design, biomedical
investigators are encouraged to postulate a nonspecific alternative hypothesis. That is,
they look to two-sided p values rather than one-sided. We concur with this approach.
Wilcoxon described his rank-order test as based on exact permutation, but permutation
of differences between mean-ranks (or, exactly equivalent, rank-sums) so as to reduce
the computational difficulties presented by R.A. Fisher's permutation procedures for
differences between means (Wilcoxon 1945). However, it is a common
misapprehension that the WMW procedure tests for equality of group medians. This is
wrong. It tests for equality of group mean ranks and, because of the ranking system
employed, mean ranks do not correspond to medians.
2.2 Variants of the WMW Procedure
There are at least three different situations with two options each which have to be
distinguished when calculating the distribution of the WMW test statistic. These were
described lucidly by Siegel and Castellan (1988) and in greater depth by Lehmann
(1998).
*Large-sample (asymptotic) approximation: (normal or x^sup 2^ distribution) versus
exact form (exact permutation distribution)
*Continuity: large sample approximation with or without correction for continuity
*Ties: large sample approximation with or without correction for ties
Exact form. In most cases the exact permutation form of WMW test should be the
preferred method. This is especially so if the experimental groups are small and,
whatever their sizes, if they are constructed by randomization rather than random
sampling (see Ludbrook and Dudley 1998).
Under the assumption of no ties, the form of the exact null distribution of the Wilcoxon
rank-sum statistic depends only on the total number of observations in the two groups
(N = n1 + n 2). It is therefore relatively easy to tabulate the exact distribution for small
group sizes, say up to n = 10. If there are tied observations, however, the above no
longer holds true, and for a given value of N the exact null distribution of the Wilcoxon
rank-sum statistic depends also on the number and pattern of tied values within and
between the groups (Lehmann 1998).
Though there are published tables of exact p values for small groups-for instance, n <=
10 (Siegel and Castellan 1988; Lehmann 1998)-these are correct only if there are no
ties. Software is now available, however, with which the exact permutation version of
the WMW test, or a Monte Carlo sampled exact version, can be executed for groups of
any size and regardless of the number and pattern of ties.
Large sample (asymptotic) approximation. What should be regarded as a large sample
is quite vague. In view of the restricted tables for the exact permutation version (see
above), most investigators are accustomed to using an asymptotic approximation when
group sizes exceed 10. The latter is either based on the normal distribution (usually) or
the X^sup 2^ distribution (sometimes). The only difference is that the X^sup 2^
distribution gives an intrinsically two-sided outcome, whereas the normal distribution
can be employed in a oneor two-sided fashion.
Continuity correction. When the normal approximation is used, this correction allows for
the fact that the normal distribution is continuous whereas the distribution of the WMW
statistic is discrete (see Siegel and Castellan 1988). This correction reduces the value
of the numerator of the z statistic and therefore renders the outcome more conservative
(i.e., gives larger p values).
Ties and corrections for ties. When observations made on an interval scale are
transformed into their corresponding ranks, equal (tied) values are assigned the mean
rank across the tie. When ties occur only within the first or within the second group, they
do not affect the outcome of the largesample approximation. However, when there is at
least one observation in the first group and at least one from the second that share a
common rank, the asymptotic version of the WMW procedure is rendered too
conservative. A correction for ties reduces the value of the denominator of the z
statistic, and so renders the outcome of the WMW procedure less conservative (i.e.,
gives smaller p values) (Siegel and Castellan 1988).
3. THE EXPERIMENT
3.1 Experimental Protocol
Animals. Male rats (strain: Sprague Dawley; BRL, Switzerland; body weight: 100-140
grams) were housed in groups of four animals (type IV cages) in a
temperaturecontrolled room under artificial illumination (6:00 a.m.-6:00 p.m., lights on)
with free access to water and food.
Rotarod test. The rotarod apparatus consists of a rotating cylinder, which is divided into
four available rat positions, each six centimeters (cm) in diameter. The cylinder is
rotated at a speed of 12 rotations per minute (rpm). The rats are placed singly on the
cylinder. One day before the experiment the animals were trained to stay on the rotarod
for 300 seconds. Rats that failed to learn the test were excluded from the study. During
the test phase the length of time each rat remains on the cylinder (the endurance time)
is measured, up to a maximum of 300 seconds.
Treatment. The animals received a fixed oral dose of a centrally acting muscle relaxant
(treatment) or a saline solvent (control). In each experiment, 24 rats were assigned to
control or treated groups by restricted randomization. The rotarod endurance time was
measured three hours after administration of the active agent or saline, because the
maximal effect of the compound was observed at this timepoint.
3.2 Results of the Experiment
These are displayed in Table 1. It is obvious that the data resulting from the experiment
could not be analyzed by the Student's t test, whatever transformation were to be
employed. It was decided, therefore, to use the WMW procedure to test whether the
mean ranks of the two groups were equal. Note that whereas the mean values and
mean ranks for endurance times of the two groups are different, the group medians are
identical.
4. RESULTS OF APPLYING THE WMW TEST
Originally, the analysis was done using SigmaStat 2.03
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Table 1.
(SPSS Inc., Chicago) and then compared with the results from SYSTAT 9 (SPSS Inc.,
Chicago). Because the outcomes were entirely different, the analysis was repeated with
the following packages: JMP 3.2.5 (SAS Institute Inc., Cary, NQ; S-Plus 2000
(MathSoft, Inc., Seattle); STATISTICA 99 Ed. Rel. 5.5 (StatSoft, Inc., Tulsa, OK);
UNISTAT 4.53b (Unistat Ltd., London); SPSS 8.0 (SPSS Inc., Chicago); Arcus
Quickstat, Biomedical Version 1.2 (Research Solutions, Cambridge, UK); Stata 6.0
(Stata Corporation, College Station, TX); and SAS 6.12 (SAS Institute Inc., Cary, NQ.
StatXact 4.0 (Cytel Software Corporation, Cambridge, MA) was used to perform the
WMW test by exact permutation, the outcome of which was used as the ultimate
benchmark of accuracy. However, before embarking on the comparative study of
statistical packages, we carried out the WMW test on the experimental data, using the
normal approximation with and without corrrections for ties and continuity, performing
the ranking by hand, using a hand-held calculator to execute the various formulations
given by Siegel and Castellan ( 1988), and referring the z statistic to computer tables of
the normal distribution (StaTable 1.0, Cytel Software Corporation, Cambridge MA). The
results of the study are summarized in Tables 2 and 3.
Calculation by hand: The results were as follows, sum of ranks 120, total rank sum 300:
SigmaStat: The medians were printed for both groups as the essential parameter under
investigation, accompanied by a P value of .085. The following text was displayed to
interpret the findings: "The differences in the median values among the two groups are
not great enough to exclude the possibility that the difference is due to random sampling
variability; there is not a statistically significant difference (p = 0.085)".
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Comment. There is no reference in the Help file or manual about the method used to
execute the test. Moreover p = .085 does not correspond to any of the hand-worked
outcomes.
SYSTAT: The WMW test is handled as a special, twogroup case of the Kruskall-Wallis
test. The output included the rank sum for each group, the Mann-Whitney U statistic =
42, the chi-square approximation = 5.948 with 1 df, and a corresponding probability of
.01473.
Comment. There is no mention in the Help file or manual about corrections for ties or
continuity. We infer from the hand-worked outcomes that a correction for ties, but not for
continuity, was used.
JMP: The output included the rank sum and rank mean for each group (denoted as
score sum and mean), a "2sample normal approximation" for the rank sum z = 2.398, p
= .0165 and a "1-way Test Chi-Square approximation" of 5.948 with 1 df and probability
of .0147.
Comment. It is not explained why the outcomes of using the normal and chi-squared
approximations are different. We infer from the hand-worked results that p = .0165 was
obtained by making corrections for both continuity and ties, p = .147 by making a
correction only for ties. S-Plus: Two options are provided in the starting window for the
WMW test: "Continuity Correction" and "Use Exact Distribution".
If "Continuity Correction" was selected, the output was: rank-sum normal statistic
without correction z = 2.4389, p-value = .0147; rank-sum normal statistic with correction
z = 2.3983, p-value = .0165.
Comment. On the basis of the hand-worked outcomes it appears that the p values are
also corrected for ties, but this is not mentioned explicitly in the output. However, a
detailed description of how the WMW procedure is implemented is given in the Help file.
If "Use Exact Distribution" was selected, the "Warning messages: cannot compute exact
p value with ties in: wil.rank.sum(x, y, alternative, exact, correct)" appeared which may
mean that the Dineen-Blakesley (1973) algorithm is used.
Comment. This implies that the S-PLUS algorithm is less versatile in performing exact
permutation than that of StatXact. STATISTICA: The rank sum for each group was
given, and as the primary result z = 1.732 and p = .083274 was reported. Then an
"adjusted" z = 2.4389 with p = .014737 is displayed. Finally, an "exact" p = .088734 is
provided.
Comment. In the Help menu it is explained that the "adjusted" z statistic was adjusted
for ties, and that for small group sizes exact probabilities are "based on the enumeration
of all possible values of the Mann-Whitney U statistic (unadjusted for ties), given the
number of observations in the two samples (see Dineen and Blakesley 1973)". Later
this is paraphrased as: "To reiterate, the computations for this probability value (for
small to moderate sized samples) are based on the assumption of no ties in the data
(ranks). Note that this limitation usually leads to only a small underestimation of the
statistical significance of the respective effects (see Siegel 1956)". Our example shows
that the "underestimation of the statistical significance" can be considerable. Nothing is
mentioned about continuity correction. From the hand-worked output, it appears that the
three p values are, respectively, the outcome with no corrections, the outcome after a
correction for ties, and the outcome of a simplified "exact" algorithm that is valid only
when there are no ties.
UNISTAT: The output included group rank sums and mean ranks, the WMW test
statistic and the statistic labeled as corrected for ties, z = -2.4389 with p = .0147, and an
"exact" probability of 0.0887. "Difference between Medians = 0" and a 95% confidence
interval for differences between medians based on normal approximation was reported
as 0 to 137.
Comment. In the Help file it is stated that for a total sample size N < 30 an "exact"
significance level is reported (referenced as employing the algorithm by Dineen and
Blakesley 1973, though this cannot cope with ties).
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Table 2.
SPSS: The output included rank sums and mean ranks for each group, the WMW test
statistic with an associated z = -2.439 and a corresponding asymptotic significance level
of p = .015. An "exact" significance probability of .089, marked as not corrected for ties,
was also displayed.
There is now available an optional module "SPSS Exact Tests", which was developed
by the Cytel Software Corporation, the vendor of StatXact, so that the statistical
methods provided are very similar.
If the "SPSS Exact Tests" module is included in your SPSS license you have an option;
"exact", and if it was chosen an additional exact (2-tailed) p = .037 is reported.
Comment. The Help menu describes the Mann-Whitney test as: "A nonparametric
equivalent to the t test. Tests whether two independent samples are from the same
population" and that "the average rank (is) assigned in the case of ties:' But nothing is
mentioned about corrections for continuity or exact probabilities. From the hand-worked
outcomes, it appears that the first p value results from a correction for ties but not
continuity. The second, "exact" p value presumably results from using the Dineen and
Blakesley (1973) algorithm, even though this is invalid when there are ties. However,
the SPSS Exact Tests module provides the correct outcome.
Arcus Quickstat: The group medians and rank sums, together with the WMW statistic,
were displayed. Exact probabilities (one- and two-sided and adjusted for ties) were
reported as .0186 and .0373, and a "95.5% confidence interval for differences between
medians or means" was reported as 0 to 137 with a "Median difference = 0".
Comment. The Help file explains that the sampling distribution of the WMW statistic is
used to calculate exact probabilities and that this can take a long time if there are tied
data. Nevertheless, the "exact" probability does result from listing the permutation
distribution and so is genuinely exact. It is not indicated how the confidence interval is
arrived at, and it contradicts the p values since it includes zero.
Stata: For the WMW test, the rank sums and the "adjusted for ties variance" are
reported, and the outcome as z = 2.439 and p = .0147. If the Kruskal-Wallis procedure
is used for two groups, a "chi-squared = 3.000 with 1 d.f., p = .0833" is reported.
Comment. There is nothing in the Help file about the statistical algorithm used in the
WMW test. In the manuals the normal approximation is described, which includes the
case that there are ties. With reference to the KruskalWallis procedure, under Methods
and Formulas in the manual it is stated that "Tied values are assigned the average
ranks." However, to judge from the hand-worked outcomes, the p-value resulting from
the chi-square approximation is not corrected for ties or for continuity. In the manual, the
general description of the WMW procedure says correctly that the hypothesis is tested
"that two independent samples . . . are from populations with the same distribution . . ."
but the example given describes (incorrectly) the outcome of the WMW procedure as
that "The results indicate that the medians are not statistically different at . . .".
SAS 6.12: PROC NPAR1WAY reports the mean ranks (denoted as scores) and a
message that "Average Scores Were Used for Ties". Then the "normal approximation
(with continuity correction of 0.5)" is given with z = 2.39826, p = 0.0165. The "Kruskal-
Wallis Test (Chi-Square Approximation)" with p = 0.0147 is also printed. In PROC
NPAR1WAY there is also available an "exact" option, which gave a two-sided exact pvalue of .0373.
Furthermore, there is an additional procedure, PROCSTATXACT, provided by the Cytel
Software Corporation, which is equivalent to StatXact and also results in p = .0373.
Comment. The user may be confused by the many options provided in SAS 6.12, even
though these are fully documented in the comprehensive manuals. Note that the "exact"
option in PROC NPAR 1 WAY and PROC-STATXACT give identical outcomes, both
being based on the permutation distribution.
StatXact: Our dataset was sufficiently small to allow the exact permutation procedure to
be followed, rather than a Monte Carlo sampled permutation distribution. Exact, oneand
two-sided, inferences were reported as p = .0186 and p = .0373, respectively. The
asymptotic outcome reports the WMW statistic, a standardized z value of 2.439, and a
two-sided p = .0147.
Comment. In the Help file there is little information about the statistical methods, but
these are described in fine detail in the manual. By reference to the hand-worked
outcomes, a correction for ties is employed in the asymptotic outcome, though this is
not made clear in the manual.
Modification of the Dataset: We repeated the calculation for some modifications of our
original dataset to illustrate the influence of ties:
* Changing one observation in the control group from 300 to 299: SYSTAT calculated
an asymptotic probability of .044 corrected for ties but not for continuity. StatXact
reported an exact p value of .0373 which is identical to the value for the unchanged
data. This shows very well how much the large sample approximation is influenced by
ties and the advantage of the exact procedure.
* The seven "300" values of the original data in the treatment group were modified
slightly to 297, 298, 299, 301, 302, 303, 304 so that no ties are present: SYSTAT gave
an asymptotic probability corrected for ties but not for continuity of .13867 and StatXact
an exact p = .1461.
5. GENERAL COMMENTARY
It seemed to us that the WMW procedure was ideally suited to the analysis of our
somewhat unusual dataset (Table 1 ). But from the point of view of nonstatisticians (the
majority of bioscientists and biomedical investigators), the results of our empirical study
are quite dismaying.
The WMW procedure tests for equality of group meanranks, not of group medians. This
is evident from our experimental data (Table 1). However, by providing group medians
or their difference in their outputs, statistics packages such as SigmaStat, Unistat,
Stata, and even Arcus QuickStat may mislead investigators into supposing that the p
values refer to the hypothesis that group medians are equal. This common
misapprehension is not unique to statistics packages. It appears in Siegel and Castellan
(1988) and many other elementary texts on statistics.
On theoretical grounds, it is clear that the only infallible way of executing the WMW test
is to compile the null distribution of the rank-sum statistic by exact permutation. This
was, in effect, Wilcoxon's ( 1945) thesis and it provided the theoretical basis for his test.
The specialized statistics package, StatXact, executed the WMW procedure in this way.
Of the general packages we reviewed, Arcus Quickstat and SAS executed the WMW
test in this way and, in the more recent versions of SAS and SPSS, modules are
available that are based on StatXact and execute the WMW test by exact permutation.
In all these cases, the two-sided outcome was p = .0373. To change tack, in the case of
our experiment, the exact permutation procedure for equality of group means also
resulted in p = .0373 (StatXact), though this is not always so. It is up to investigators to
decide whether a test for equality of group mean-ranks (but not of group medians) is
more informative than one for equality of group means.
Three packages claimed to execute the WMW procedure in an "exact" fashion:
STATISTICA, UNISTAT, and SPSS. In each case the result was p = .0887 (or .089).
The packages refer to Dineen and Blakesley (1973) for the algorithm used to calculate
their "exact" form of the WMW. A closer look at calculating the exact distribution
(Lehmann 1998) shows that this algorithm relies only on the sizes of the two groups,
which is only correct for untied data since the number of ties is not taken into
consideration. In our view, to output such "exact" p values in the obvious case of tied
data is dangerously misleading and results in no more than pseudo-exact outcomes. It
should also be noted that published tables of exact outcomes of the WMW procedure
(Siegel and Castellan 1988; Lehmann 1998) are invalid when ties are present.
The several p values provided within the packages are likely to confuse rather than
instruct the biomedical investigator (and even the unwary statistician), especially since
the formulations of the WMW test which result in the different p values are not clearly
defined. Scientists tend to look for "significant" results from their experiments, so that
some may be inclined to select p = .0147 (which results from using the normal
approximation with a correction for ties but not for continuity).
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Table 3.
A survey of the type of p values produced by all the reviewed packages is given in
Table 2.
6. CONCLUSIONS
We summarize our investigation in the following points and conclude with some
recommendations.
1. In general, microcomputer statistics packages provide very inadequate
documentation in their manuals and Help files of precisely how the WMW test is
executed (see Table 3). As a consequence, the results can be dangerously misleading.
It is essential that explicit documentation be given. The user must be in no doubt about
which formulations of the WMW test are used.
2. The output of the results should be clear, fully explained, and comprehensible to
nonstatisticians.
3. Different microcomputer statistics packages can give very different outcomes for the
WMW test (see Table 2).
4. Investigators cannot rely on the popular, generalpurpose, microcomputer statistics
programs which are reviewed here to provide an accurate outcome from the WMW test.
This is because the programs usually use one or more versions of the large-sample
(asymptotic) approximation. There are exceptions to this statement: Arcus Quickstat,
SAS, and also additional new modules for SPSS and SAS, execute the test by exact
permutation. The specialized package StatXact always uses exact permutation.
5. If investigators use a statistics package that we have not reviewed here to execute
the WMW test, we strongly recommend they should in the first instance analyze their
data by hand, or use the example we give here in Table 1, to establish which variant of
the test is executed (see Table 2).
6. If the original data are in ranked form, the WMW procedure is the best available,
provided the test is executed by exact permutation (e.g., StatXact, Arcus Quickstat,
SAS and special modules in SAS and SPSS).
7. If the original data are in continuous (interval scale) form, but are clearly non-normally
distributed or have been acquired by randomization rather than random sampling, a
permutation (randomization) test for equality of group means may be a better option
than the WMW test for equality of mean ranks. This can be executed by StatXact.
[Received March 1999. Revised October 1999.]
[Reference]
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Ludbrook, J., and Dudley, H. (1998), "Why Permutation Tests are Superior to t and F Tests in Biomedical Research," The
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Mann, H. B., and Whitney, D. R. (1947), "On a Test of Whether One of Two Random Variables is Stochastically Larger than
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[Author note]
Reinhard Bergmann and Will Spooren are Scientists with Novartis Pharma, Department of Research, P.O. Box CH-4002
Basel, Switzerland (E-mail: reinhard.bergmann@pharma.Novartis.com). John Ludbrook is Professorial Fellow with the
University of Melbourne, Department of Surgery, Royal Melbourne Hospital, Parkville, Victoria 3050, Australia. The authors
thank SPSS, Inc., Switzerland (SPSS 8.0); Research Solutions, Cambridge, UK (Arcus Quickstat, Biomedical Version 1.2);
and Stata Corporation, College Station, TX, (Stata 6.0) for providing copies of the software for evaluation.